Pure jump process in Duffie, Pan and Singleton's paper

Question

In page 1349 or Section 2.1 of "Duffie, D., Pan, J., & Singleton, K. (2000). Transform Analysis and Asset Pricing for Affine Jump-Diffusions. Econometrica, 68(6), 1343-1376" the pure jump process Z is defined as follow:

conditional on the path of X, the jump times of Z are the jump times of a Poisson process with time-varying intensity $\{\lambda(X_s):0\le{s}\le{t}\}$, and that the size of the jump of Z at a jump time T is independent of $\{X_s:0\le{s}<T\}$ and has the probability distribution $\nu$.

Since I can't find a formal definition of Z in the same paper, I'm writing to ask if it is talking about a compound Poisson Process. Saying, $(Z_t)_{t\ge{0}}$ is a stochastic process such that: $Z_t=\sum_{i=1}^{N_t}{Y_i}$, where $Y_1,...,Y_{N_t}$ are random variables with distribution $\nu$ and $N_t$ follows a $Pois(\lambda(X_t))$.

Answer 1

Essentially yes - $Z_t$ is a compound Poisson process, except that the underlying counting process $N_t$ has intensity $\lambda(X_t)$. I.e $$ N_t - N_s \sim Pois\bigg( \int_s^t \lambda(X_u) \mathrm{d}u\bigg). $$

In Lewis (2001), he refers to the 'jump diffusion-case', when the underlying process is a Levy process with finite Levy measure - Which is consistent with the above.

Lewis, Alan L., A Simple Option Formula for General Jump-Diffusion and Other Exponential Levy Processes (September 2001). Available at SSRN: https://ssrn.com/abstract=282110 or http://dx.doi.org/10.2139/ssrn.282110